SO(3) Special Othorgonal Group

In summary, for the special orthogonal group SO(3) and G-set R^3, the stabilizer of a chosen point x in R^3 (not equal to 0) is all the rotations about the axis produced by x (and -x). This is isomorphic to SO(2), the set of rotations in the plane orthogonal to x. The geometric interpretation for SO(2) is the group of all rotations by any angle in R^2. It may not be obvious at first, but this is how it is defined in the notes.
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For the special orthogonal group SO(3), with G-set R^3, and the usual G-action, we choose x in R^3 not equal to 0. Then the stabilizer of x (set of all the transformations in SO(3) that doesn't change x) is all the rotations about the axis produced by x (and -x). Can someone explain why the stabilizer of x is isomorphic to SO(2)? The notes I have just writes this as if it should be obvious. Am I missing something? Also, just to make sure I'm understanding this right, is the geometric interpretation for SO(2) the group of all rotations by any angle in R^2?
 
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The set of rotations around an axis x in R^3 is isomorphic to the set of rotations in the plane orthogonal to x. And, yes, SO(2) is the set of rotations of a plane.
 

What is SO(3) Special Orthogonal Group?

The SO(3) Special Orthogonal Group, also known as the Rotation Group in three-dimensional Euclidean space, is a mathematical group that represents all possible rotations in three-dimensional space. It is a fundamental concept in linear algebra and has a variety of applications in physics, computer graphics, and robotics.

How is SO(3) Special Orthogonal Group represented?

In mathematics, SO(3) is typically represented by a matrix of size 3x3, where each row and column represents a unit vector in three-dimensional space. This matrix is known as a rotation matrix and is used to rotate a vector or object by a specified angle around a given axis.

What are the properties of SO(3) Special Orthogonal Group?

SO(3) Special Orthogonal Group has several important properties, including closure, associativity, and the existence of an identity element (the identity matrix). It is also a non-abelian group, which means that the order of operations matters, and its elements do not commute.

What is the relationship between SO(3) Special Orthogonal Group and other mathematical concepts?

SO(3) Special Orthogonal Group is closely related to other mathematical concepts, such as quaternions, rotation vectors, and Euler angles. It is also a subgroup of the Special Linear Group, which is a group of all invertible matrices with determinant equal to 1.

What are the applications of SO(3) Special Orthogonal Group?

SO(3) Special Orthogonal Group has a wide range of applications in various fields, including robotics, computer graphics, and physics. It is used to model and control the motion of rigid bodies, such as robots and spacecraft, and to create realistic 3D graphics in video games and animation. It also plays a crucial role in describing the rotational symmetries of physical systems in quantum mechanics and classical mechanics.

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